We will now study the symmetries and defects introduced in Sections REF  and REF 
using the lattice formulation of two-dimensional Yang-Mills theory, as
reviewed for example in [1]}, [2]}.
To discretise the calculation
we fix a cell decomposition consisting of an embedded graph which covers the compact oriented surface \(\Sigma \)  into vertices \(\Sigma ^{\textrm {\tiny (0)}}\) , edges \(\Sigma ^{\textrm {\tiny (1)}}\) ,
and faces \(\Sigma ^{\textrm {\tiny (2)}}\) . As the faces are contractible, the only remainders of a principal \(G\) -bundle on
\(\Sigma \)  are
its fibre over every vertex, which we can trivialise. A gauge transformation
is therefore described by a map \(\xi :\Sigma ^{\textrm {\tiny (0)}}\longrightarrow G\) . A connection on the
bundle is described by its parallel transport, which is a group element \(g_\gamma \in G\) 
for every edge \(\gamma \in \Sigma ^{\textrm {\tiny (1)}}\) ; if the edge \(\gamma \) 
joins vertex \(x\)  to vertex \(y\) , then a gauge transformation \(\xi \)  acts on
\(g_\gamma \)  by \(g_\gamma \longmapsto \xi _y\,g_\gamma \,\xi _x^{-1}\) . The
curvature of a connection is a gauge-invariant map
\(\mathcal {U}:\Sigma ^{\textrm {\tiny (2)}} \longrightarrow G\) .
In the lattice formulation the only gauge-invariant quantity one can
construct for a face \(w\in \Sigma ^{\textrm {\tiny (2)}}\)  is the (conjugacy class of the) holonomy
around the face:
\(\mathcal {U}_w= \prod _{\gamma \in \partial w}\, g_\gamma \ ,\) 
